Stabilized Space Time Fluid Structure Interaction (SSTFSI) Technique Wind Fabric Interactions
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1 Stabilized Space Time Fluid Structure Interaction (SSTFSI) Technique Wind Fabric Interactions Tayfun E. Tezduyar, Sunil Sathe and Jason Pausewang Mechanical Engineering, Rice University MS Main Street, Houston, Texas 77005, USA {tezduyar, sathe, Summary. The stabilized space time fluid structure interaction (SSTFSI) techniques [1] developed by the Team for Advanced Flow Simulation and Modeling (T AFSM) were applied to a number of 3D examples in [1] and additional examples in [2, 3], including arterial fluid mechanics [2]. Here we focus on a subset of those applications wind fabric interactions. We describe the additional numerical challenges involved in this class of applications and the supplementary techniques developed to address those challenges. The supplementary techniques include the FSI Geometric Smoothing Technique (FSI-GST) and using split nodal values for pressure at the edges of the fabric and incompatible meshes at the air fabric interfaces. With the FSI-GST, the fluid mechanics mesh is sheltered from the consequences of the geometric complexity of the structure. Using split nodal values for pressure at the edges and incompatible meshes at the interfaces stabilizes the structural response at the edges of the membrane used in modeling the fabric. As two test cases, we compute the wind fabric interactions of a windsock and a pair of sails. 1 Introduction The stabilized space time FSI (SSTFSI) techniques were recently introduced in [1] to increase the scope and performance of the earlier versions of the space time FSI techniques developed by the Team for Advanced Flow Simulation and Modeling (T AFSM). The stabilization methods used in these techniques are the Streamline-Upwind/Petrov-Galerkin (SUPG) [4] and Pressure- Stabilizing/Petrov-Galerkin (PSPG) [5] formulations. A number of 3D numerical examples computed with the SSTFSI techniques were also presented in [1], with additional numerical examples presented in [2, 3], including numerical examples from arterial fluid mechanics [2]. In this paper, we focus on FSI computation of a subset of those applications wind fabric interactions. The versions of the SSTFSI technique we use in the computations presented here are the SSTFSI-TIP1, which was described in Remarks 5 and 10 in [1], and SSTFSI-SV, which was described in Remarks 6 and 10. The fabric is modeled with the membrane element, which does not offer bending stiffness, and is assumed to be made of linearly elastic material. Wind fabric interactions pose a number of numerical challenges beyond those posed by an FSI problem in general. These additional challenges include sheltering the fluid mechanics mesh from the consequences of the geometric complexity of the structure and stabilizing the structural response at the edges of the membrane structures without the benefit of any bending stiffness. For computations where the geometric complexity of the structure would require a fluid mechanics mesh that is not affordable or not desirable or just not manageable in mesh moving, we proposed in [1] the FSI Geometric Smoothing Technique (FSI-GST). In this technique, the structural mesh and displacement rates at the interface are projected to the fluid mesh after a geometric smoothing. In the geometric smoothing, a value (mesh coordinate or displacement rate) at a given node is replaced by a weighted average of the values at that node and a limited set of nearby nodes. When projecting the stress values from the smoothened interface to the
2 structure, currently we just transfer those values to the corresponding nodes of the structure. In some computations, we may need not an isotropic geometric smoothing but a directional smoothing along some preferred direction. For such computations, we proposed in [1], as a version of the FSI-GST, the FSI Directional Geometric Smoothing Technique (FSI-DGST). In the FSI-DGST, whenever we can, we generate the interface mesh in such a fashion that the preferred smoothing directions can approximately be represented by the gridlines of the interface mesh. Then the weighted averaging for a node on such a gridline would involve a limited set of nearby nodes only along that gridline. The directional smoothing concept is similar to the directional upwind concept of the SUPG formulation, where the residual-based numerical dissipation is active only along the streamline direction. In our computations the fabric is modeled with the membrane element, which does not offer bending stiffness, and this can lead to excessive bending at the edges. Two of the supplementary techniques we use in conjunction with the SSTFSI techniques provide some solution to this issue. The first one, proposed in Remark 9 in [1], is using split nodal values for pressure not only in the interiors but also at the boundaries (i.e. edges) of a membrane structure submerged in the fluid. Our earlier experience shows that this provides additional numerical stability for the edges of the membrane. The second one, proposed in [3], is using incompatible meshes at the fluid structure interface as a means for limiting the excessive bending to narrow regions near the edges. This is accomplished by increasing the structural-mesh refinement near the edges without an equal refinement increase for the fluid mesh in the same regions of the fluid structure interface. 2 Test computations 2.1 Windsock Figure 1 shows a windsock from Albuquerque airport. The windsock model in our computation Figure 1: A windsock. has a length of 1.5 m and a diameter ranging from 0.25 m upstream to 0.15 m downstream (see Figure 2). Initially the windsock is in a horizontal position, and the starting condition for the 10 m/s Rigid 0.25 m 0.15 m 1.5 m Figure 2: Windsock. Problem setup.
3 flow field is the developed flow field corresponding to a rigid windsock held in that horizontal position. Then the gravity is turned on for the windsock, the FSI starts, and the windsock starts bending down. The wind velocity is constant at 10 m/s. The air density and kinematic viscosity are set to 1.2 kg/m 3 and m 2 /s. The thickness, density, stiffness and Poisson s ratio for the windsock are 2.0 mm, 100 kg/m 3, N/m 2 and 0.45, respectively. The upstream edge of the structure is held fixed while the remaining structure is free and flaps in cycles. The mesh for the windsock is semi-structured and consists of 984 nodes and 1,920 three-node triangular membrane elements. The fluid mechanics mesh contains 19,579 nodes and 113,245 four-node tetrahedral elements. Initially, the fluid mesh at the interface is identical to the windsock mesh. The computation is carried out with the SSTFSI-SV technique (see Remarks 6 and 10 in [1]) and the SUPG test function option WTSE (see Remark 2 in [1]). The stabilization parameters used are those given in [1] by Eqs. (9) (12) and (14) (17). The GMRES search technique is used with a diagonal preconditioner. The time-step size is s, and the computation duration is two cycles of flapping. The number of nonlinear iterations per time step is 5, and the number of GMRES iterations per nonlinear iteration is 30. We expected the windsock to develop kinks as it blows in the wind. Therefore we used the FSI-GST for smoothing the fluid mesh at the interface. The nodes of the windsock mesh were generated on straight longitudinal gridlines, and with that we were able to use the directional version of the FSI-GST, i.e. the FSI-DGST. For a node on such a gridline, we use a weighted averaging involving four nearby nodes on each side (for details, see Section 12.4 in [1]). We note that this directional smoothing does not introduce any smoothing in the circumferential direction. During the FSI computations the structure develops kinks, which would make updating the fluid mechanics mesh more difficult and increase the frequency of remeshing. With the FSI-DGST, two cycles of flapping were computed without any remeshing. Figure 3 shows the structural and fluid mechanics meshes at the interface, one with a kink and the other one smooth. Figure 4 shows the windsock and the flow field at various instants. Figure 3: Windsock. Meshes at the interface: structure (left) and fluid (right). 2.2 Sails Figure 5 shows the dual-sail configuration. The mainsail geometry and initial shape are derived from a plate airfoil design with maximum camber of 15% cord length located at 30% cord. This is an approximation to the mainsail of the Adventuress [6]. The dimensions are 15 m along the mast and 5 m along the foot. Mainsail nodes are fixed along the luff (leading edge) and foot and free elsewhere. A single jib sail was proportionally generated from the same photo. The jib is 12.7 m along the luff. All nodes of the jib are free, with the head (top corner), tack
4 Figure 4: The windsock and the flow field (velocity and pressure) at various instants. Velocity vectors colored by magnitude. (front corner), and clew (aft corner) attached to cables with fixed ends. The airflow velocity is 7.72 m/s (approximately 10 knots) at an angle of 35 from the centerline of the boat (not modeled). The air density and kinematic viscosity are set to 1.2 kg/m 3 and m 2 /s. The thickness, density, stiffness and Poisson s ratio for the sails are 1.0 mm, 1,370 kg/m 3, N/m 2 and 0.3, respectively. The diameter, density, stiffness and Poisson s ratio for the cables are 6.35 mm, 1,440 kg/m 3, N/m 2 and 0.3, respectively. Figure 5 also shows the fluid mechanics mesh at the interface and the sail meshes. The mesh for the mainsail consists of 654 nodes and 1,177 three-node triangular membrane elements. The mesh for the jib sail has 349 nodes and 601 elements. The mesh for the cables consist of 3 two-node cable elements. The fluid mechanics mesh contains 91,512 nodes and 553,183 four-node tetrahedral elements, with 558 nodes and 952 triangular faces at the fluid structure interface. The computation is carried out with the SSTFSI-TIP1 technique (see Remarks 5 and 10 in [1]) and the SUPG test function option WTSA (see Remark 2 in [1]). The stabilization parameters used are those given in [1] by Eqs. (9) (12), (14) (15) and (17). In Eq. (14), the τ SUGN2 term is dropped. The GMRES search technique is used with a diagonal preconditioner. The time-step size at the beginning of the computation is s, and is ramped up in the later stages to s, s, and s. The number of nonlinear iterations per time step is 5, and the number of GMRES iterations per nonlinear iteration is 30. Figure 6 shows the sails at various instants. 3 Concluding remarks In this paper we focused on the application of the stabilized space time fluid structure interaction (SSTFSI) techniques to wind fabric interactions. In addition to the typical numerical challenges involved in an FSI problem, wind fabric interactions pose some specific challenges. In some cases, the geometric complexity of the structure might require a fluid mechanics mesh
5 Figure 5: Sails. Configuration and dimensions (left), fluid mechanics mesh at the interface (middle), and sail meshes (right). that is not affordable or not desirable or just not manageable in mesh moving. To avoid that, as a supplementary technique, we use the FSI Geometric Smoothing Technique (FSI-GST). In this approach, the structural mesh and displacement rates at the interface are projected to the fluid mechanics mesh after a geometric smoothing, thus not passing to the fluid mechanics mesh the unresolvable modes of the fabric deformation. More specifically, we use the directional version of the FSI-GST the FSI Directional GST (FSI-DGST), where the smoothing is applied along the direction it is needed. The directional smoothing concept is similar to the directional upwind concept of the SUPG formulation, where the residual-based numerical dissipation is active only along the streamline direction. Because the fabric in our computations is modeled with the membrane element, which does not offer bending stiffness, in some cases we might have excessive bending at the edges of the fabric. To remedy that, we use split nodal values for pressure not only in the interiors but also at the edges of membrane structures. This stabilizes the structural response at the edges of the membrane. As an additional remedy, we use incompatible meshes at the fluid structure interface as a means for limiting the excessive bending to narrow regions near the edges. Increasing the structural-mesh refinement near the edges without an equal refinement increase for the fluid mesh in the same regions accomplishes this goal. As test cases, we computed the wind fabric interactions of a windsock and a pair of sails. The results demonstrate that the SSTFSI technique, together with the supplementary techniques we described, can successfully address the numerical challenges involved in wind fabric interactions. This work was supported in part by NASA Johnson Space Center under Grant NNJ06HG84G and by the Rice Computational Research Cluster funded by NSF under Grant CNS , and a partnership between Rice University, AMD and Cray. References [1] T.E. Tezduyar and S. Sathe, Modeling of fluid structure interactions with the space time finite elements: Solution techniques, International Journal for Numerical Methods in Fluids, published online, DOI: /fld.1430, January 2007.
6 Figure 6: Sails at various instants. [2] T.E. Tezduyar, S. Sathe, T. Cragin, B. Nanna, B.S. Conklin, J. Pausewang, and M. Schwaab, Modeling of fluid structure interactions with the space time finite elements: Arterial fluid mechanics, International Journal for Numerical Methods in Fluids, published online, DOI: /fld.1443, February [3] T.E. Tezduyar, J. Pausewang, and S. Sathe, FSI modeling of sails, in P. Bergan, J. Garcia, E. Onate, and T. Kvamsdal, editors, Marine 2007, CIMNE, Barcelona, Spain, (2007). [4] A.N. Brooks and T.J.R. Hughes, Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 32 (1982) [5] T.E. Tezduyar, Stabilized finite element formulations for incompressible flow computations, Advances in Applied Mechanics, 28 (1992) [6]
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